Question

convert the rectangular coordinates to polar coordinates with r > 0 and 0 ≤ θ < 2π. (0, -√7) (r, θ) = ( )

Explanation:

Step1: Calculate \( r \)

The formula for \( r \) in polar coordinates from rectangular coordinates \((x, y)\) is \( r=\sqrt{x^{2}+y^{2}} \). Here, \( x = 0 \) and \( y=-\sqrt{7} \). So, \( r=\sqrt{0^{2}+(-\sqrt{7})^{2}}=\sqrt{0 + 7}=\sqrt{7} \) (since \( r>0 \)).

Step2: Calculate \( \theta \)

The formula for \( \theta \) is \( \tan\theta=\frac{y}{x} \), but when \( x = 0 \), we look at the position of the point. The point \((0,-\sqrt{7})\) is on the negative \( y \)-axis. In the range \( 0\leq\theta<2\pi \), the angle for the negative \( y \)-axis is \( \frac{3\pi}{2} \).

Answer:

\((\sqrt{7},\frac{3\pi}{2})\)